Find remainder when $ x^{3}-x^{2}-44x+91 $ is divided by $ x-7 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&1&-1&-44&91\\& & 7& 42& \color{black}{-14} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{-2}&\color{orangered}{77} \end{array} $$The remainder when $ x^{3}-x^{2}-44x+91 $ is divided by $ x-7 $ is $ \, \color{red}{ 77 } $.
Explanation
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-1&-44&91\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 1 }&-1&-44&91\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 1 } = \color{blue}{ 7 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-1&-44&91\\& & \color{blue}{7} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 7 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}7&1&\color{orangered}{ -1 }&-44&91\\& & \color{orangered}{7} & & \\ \hline &1&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 6 } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-1&-44&91\\& & 7& \color{blue}{42} & \\ \hline &1&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -44 } + \color{orangered}{ 42 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}7&1&-1&\color{orangered}{ -44 }&91\\& & 7& \color{orangered}{42} & \\ \hline &1&6&\color{orangered}{-2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&1&-1&-44&91\\& & 7& 42& \color{blue}{-14} \\ \hline &1&6&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 91 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 77 } $
$$ \begin{array}{c|rrrr}7&1&-1&-44&\color{orangered}{ 91 }\\& & 7& 42& \color{orangered}{-14} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{-2}&\color{orangered}{77} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 77 }\right) $.